Anomalous non-additive dispersion interactions in systems of three onedimensional wires
Misquitta, AJ; Maezono, R; Drummond, ND; Stone, AJ; Needs, RJ
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PHYSICAL REVIEW B 89, 045140 (2014)
Anomalous nonadditive dispersion interactions in systems of three one-dimensional wires
Alston J. Misquitta
School of Physics and Astronomy, Queen Mary, University of London, London E1 4NS, United Kingdom
Ryo Maezono
School of Information Science, JAIST, Asahidai 1-1, Nomi, Ishikawa 923-1292, Japan
Neil D. Drummond
Department of Physics, Lancaster University, Lancaster LA1 4YB, United Kingdom
Anthony J. Stone
The University Chemical Laboratory, Lensfield Road, Cambridge CB2 1EW, United Kingdom
Richard J. Needs
TCM Group, Cavendish Laboratory, 19, J. J. Thomson Avenue, Cambridge CB3 0HE, United Kingdom
(Received 8 August 2013; revised manuscript received 9 January 2014; published 29 January 2014)
The nonadditive dispersion contribution to the binding energy of three one-dimensional (1D) wires is
investigated using wires modeled by (i) chains of hydrogen atoms and (ii) homogeneous electron gases. We
demonstrate that the nonadditive dispersion contribution to the binding energy is significantly enhanced compared
with that expected from Axilrod-Teller-Muto–type triple-dipole summations and follows a different power-law
decay with separation. The triwire nonadditive dispersion for 1D electron gases scales according to the power
law d −β , where d is the wire separation, with exponents β(rs ) smaller than 3 and slightly increasing with rs from
2.4 at rs = 1 to 2.9 at rs = 10, where rs is the density parameter of the 1D electron gas. This is in good agreement
with the exponent β = 3 suggested by the leading-order charge-flow contribution to the triwire nonadditivity, and
is a significantly slower decay than the ∼d −7 behavior that would be expected from triple-dipole summations.
DOI: 10.1103/PhysRevB.89.045140
PACS number(s): 68.65.La, 02.70.Ss
I. INTRODUCTION
Recently, there has been a resurgence in attempts to model
the dispersion interaction between low-dimensional nanoscale
objects more accurately. Using an array of electronicstructure [1–3] and analytical [4] techniques, several groups
have demonstrated that the dispersion interaction between oneand two-dimensional systems can deviate strongly from that
expected from the well-known additive picture of r −6 -type
interactions [5,6]. For the case of parallel one-dimensional
(1D) metallic wires separated by distance d, Dobson et al.
[4] demonstrated that the van der Waals dispersion interaction
should decay as ∼ − d −2 [ln(γ d)]−3/2 , where γ is a constant
that depends on the wire width. This analytic result was
subsequently verified by Drummond and Needs [3] using
diffusion quantum Monte Carlo (DMC) calculations [7]. This
change in the power law of the dispersion energy can be
understood as arising from correlations in extended plasmon
modes in the metallic wires [4,8,9]. These plasmon modes
would be expected in any low-dimensional system with a
delocalized electron density.
Misquitta et al. [2] have recently extended these results to
the more general case of finite- and infinite-length wires with
arbitrary band gap. Using dispersion models that include nonlocal charge-flow polarizabilities, they were able to describe
the dispersion interactions in all cases, including the insulating
and semimetallic wires. In these models, the plasmonlike
fluctuations are modeled by the charge-flow polarizabilities
which, at lowest order, result in a −d −2 dispersion interaction
[2,5]. For metallic wires, these terms are dominant at all
1098-0121/2014/89(4)/045140(9)
separations and yield the result of Dobson et al. for the
dispersion.
Curiously, many of these results were known as early as
1952. Using a tight-binding Hückel-type model for linear
polyenes, Coulson and Davies [10] investigated the dispersion
interactions between the chains in a variety of configurations
and with a range of highest occupied to lowest unoccupied
molecular orbital (HOMO–LUMO) gaps. Their conclusions
about the nonadditivity of the dispersion interaction and
the changes in power law (deviations from the expected
effective −d −5 London behavior) are essentially identical
to those reached by Misquitta et al. [2]. A few years later,
Longuet-Higgins and Salem [11] reached similar conclusions
and related the nonadditivity of the dispersion to the existence
of long-range correlations within the system. A decade later,
Chang et al. [12] used Lifshitz theory to derive an analytic
form of the dispersion interaction between two metallic wires
that is identical to the expression of Dobson et al. [4], although
the latter considered many more cases.
The current interest in this field stems from two sources.
First, we have recently witnessed an explosion of work
on nanoscale devices confined in one or two dimensions.
Examples are carbon nanotubes and devices based on graphene
and related materials. To model accurately the self-assembly
of these materials, we need to describe correctly their interactions, particularly the ubiquitous dispersion interaction.
Second, ab initio electronic-structure methods have now
achieved a level of accuracy and computational efficiency that
allows them to be applied to such systems. These methods have
exposed the inadequacies of assumptions and approximations
045140-1
©2014 American Physical Society
MISQUITTA, MAEZONO, DRUMMOND, STONE, AND NEEDS
PHYSICAL REVIEW B 89, 045140 (2014)
made in many empirical models. From the research cited
above, we now know that the dispersion energy exhibits much
more substantial nonadditivity than assumed previously.
We emphasize here that empirical models for the dispersion
energy prove inadequate because they rely on the assumption
6
of additivity through the pairwise C6ab /rab
model with van der
ab
Waals coefficients C6 between sites a and b assumed to be
isotropic constants, with little or no variation with changes
in chemical environment. Part of the missing nonadditivity
arises from the local chemical environment changes and from
through-space coupling between the dipole oscillators. The
remainder arises from the metalliclike contributions that are
responsible for the anomalous dispersion effects that are the
subject of this paper. We stress that while the first kind of
nonadditivity can be described by coupled-oscillator models
[13] and ab initio derived dispersion models such as those
obtained from the Williams-Stone-Misquitta [14,15] effective
local polarizability models, as we shall see next, the latter,
that is, the nonadditivity arising from metallic contributions,
requires models that take explicit account of extended charge
fluctuations.
The unusual nature of the second-order dispersion energy
(2)
Edisp
for infinite, parallel 1D wires of arbitrary band gap can
be understood as follows. The electronic fluctuations in the
wire are broadly of two types: the short-range fluctuations
associated with tightly bound electrons and the long-range
plasmon-type fluctuations associated with electrons at the band
edge. The former give rise to the standard dispersion model,
while the latter are responsible for the effects discussed in this
paper and those cited above. For systems with a finite gap,
the plasmonlike modes will be associated with a finite length
scale lc defined, for example, via the Resta localization tensor
[16]. For metallic systems, this length scale is expected to
diverge. Consider now the two cases depicted in Fig. 1. In the
first case, the wires are separated by d < lc . Here, the leadingorder contribution from the spontaneous extended fluctuation
depicted in the figure is that between charges and leads to the
(2)
: the spontaneous fluctuation at the first
−d −2 behavior of Edisp
wire results in a field ∼d −1 at the second and this interacts with
the first via another d −1 interaction leading to the favorable
−d −2 dispersion energy. Only local charge pairs contribute
to this leading-order interaction, consequently, the dispersion
interaction per unit length remains −d −2 .
If, on the other hand, d lc , the extended fluctuation at
the first wire generates a dipole field of strength ∼d −3 at the
second, and the resulting induced (extended) dipole interacts
with the first via a dipole-dipole interaction leading to another
factor of d −3 . This gives a net favorable dispersion interaction
of −d −6 . In this case, to find the net dispersion interaction
per unit wire length we need to sum over all the interactions
between an element of one wire and all elements of the other,
which leads to an effective −d −5 dispersion interaction just
as for the pointlike fluctuating dipoles of the tightly bound
electrons [[17], p. 173]. In both cases, the usual −d −5 effective
dispersion interaction from the tightly bound electrons must
be included too.
The length scale lc is expected to diverge in a metal, leading
(2)
. For finite-gap wires we
to a single power law −d −2 for Edisp
expect the two regimes described above. This is exactly the
conclusion reached by Misquitta et al. [2] and, much earlier,
by Coulson and Davies [10].
The second-order dispersion energy is, however, only part
of the story. For a group of interacting monomers (possibly of
different types), the dispersion energy includes contributions
from second-order as well as third- and higher-order terms. The
third-order dispersion includes two- and three-body terms [18];
(3)
(3)
[2] and the latter by Edisp
[3].
the former will be denoted by Edisp
FIG. 1. (Color online) Electronic fluctuations in (infinite) 1D
wires (in blue) arise from the tightly bound electrons (not shown)
and electrons at the band edge (represented by the red arrows). The
extent of these fluctuations will depend on the band gap (see text)
and will have a typical length scale lc . An extended fluctuation of
+ . . . − in one wire will induce a − . . . + fluctuation in the other. If d
is the separation, we can identify two cases: (1) d < lc and (2) d lc .
As explained in the text, the leading-order dispersion interaction in
the former is associated with charge-induced-charge interactions, and
that of the latter with dipole-induced-dipole interactions.
(3)
Edisp
[2] is expected to be important for small-gap systems
since these are associated with large hyperpolarizabilities, but
(3)
[3] decays slowly
we may expect a priori that as long as Edisp
enough with trimer separation, it is the three-body nonadditive
(3)
[3] that will be the dominant contributor in the
energy Edisp
condensed phase due to the far larger number of trimers
compared with dimers.
(3)
[3] is usually modThe three-body nonadditive energy Edisp
eled using the triple-dipole Axilrod-Teller-Muto expression
(3)
(see Sec. II) [19,20] from which Edisp
[3] ∼ R −9 , that is, the
nonadditivity decays very rapidly with separation. As will be
demonstrated in the following, this expression is not valid
for small-gap systems; instead, a more general expression is
derived that includes contributions from correlations between
the long-wavelength plasmonlike modes. From the physical
picture of the second-order dispersion energy given above, we
(3)
may a priori expect that the true Edisp
[3] will be qualitatively
different from that suggested by the triple-dipole expression.
As we shall see in the following, this is indeed the case.
The multipole expansion is a powerful method, but it would
be reassuring to verify its predictions using a nonexpanded
ab initio approach. In order to obtain hard numerical data
describing the nonadditivity of the dispersion interactions
between metallic wires, we have evaluated the binding energy
of three parallel, metallic wires in an equilateral-triangle
045140-2
ANOMALOUS NONADDITIVE DISPERSION INTERACTIONS . . .
configuration using the variational and diffusion quantum
Monte Carlo (VMC and DMC) methods. VMC allows
one to take expectation values with respect to explicitly
correlated many-electron wave functions by using a Monte
Carlo technique to evaluate the multidimensional integrals.
The DMC method projects out the ground-state component
of a trial wave function by simulating drift, diffusion, and
branching processes governed by the Schrödinger equation
in imaginary time. In our quantum Monte Carlo (QMC)
calculations, each wire was modeled as a 1D homogeneous
electron gas (HEG). The dependencies of the biwire and triwire
interactions on the wire separation d were evaluated in order
to determine the asymptotic power law for the interaction
and the nonadditive three-body contribution. We find that the
long-range nonadditivity is repulsive and scales as a power law
in d with an exponent slightly less than three.
The paper is organized as follows. The underlying theory is
described in Sec. II. In Sec. III, we describe the computational
details and present our results. Finally, we discuss the physical
consequences of our results in Sec. IV.
imaginary frequency iu [21,22]. The sign of the above
expression has been chosen so that the polarizability tensor
defined as
aa Q̂aα (r1 )α(r1 ,r1 ; ω)Q̂aα (r1 )d 3 r1 d 3 r1 (2)
ααα (ω) = −
is positive-definite. Here, Q̂aα is the multipole moment operator
for site a with component α = 00,10,11c,11s, . . . using the
aa notation described by Stone [5]. As defined, ααα
(ω) is the
distributed polarizability for sites a and a . It describes
the linear response of the expectation value of the local
operator Q̂aα to the frequency-dependent (local) perturbation
aa Q̂aα cos(ωt) [23]. That is, the distributed polarizability ααα
(ω)
describes the first-order change in multipole moment of
component α at site a in response to the frequency-dependent
perturbation of component α at a site a .
For the sake of clarity, we will use the following notation in
subsequent expressions: sites associated with monomers A, B,
and C will be designated by a,a , b,b , and c,c , and angular
momentum labels by α,α , β,β , and γ ,γ , respectively.
Molecular labels are hence redundant and will be used only if
there is the possibility of confusion.
(3)
[3] is obtained by expandThe multipole expansion of Edisp
ing the Coulomb terms in Eq. (1) as follows:
II. THEORY
The nonexpanded three-body, nonadditive dispersion energy has been shown to be [18] (all formulas will be given in
SI units, but results will be in atomic units)
(3)
Edisp
[3]
=−
π (4π 0 )3
×
1
ab b
= Q̂aα (r1 )Tαβ
Q̂β (r2 ),
|r1 − r2 |
∞
du
d 3 r1 d 3 r1 d 3 r2 d 3 r2 d 3 r3 d 3 r3
(1)
Here, α X (r1 ,r1 ; iu) is the frequency-dependent density susceptibility (FDDS) function for monomer X evaluated at
→
(3)
Edisp,MP
[3]
×
=+
T a b T b c T c a
π (4π 0 )3 α β β γ γ α
∞
3
d r1 d
0
d 3 r2 d 3 r2 Q̂bα (r2 )α B (r2 ,r2 ; iu)Q̂bα (r2 )
T a b T b c T c a
= +
π (4π 0 )3 α β β γ γ α
0
∞
3
r1 Q̂aα (r1 )α A (r1 ,r1 ; iu)Q̂aα (r1 )
d 3 r3 d 3 r3 Q̂cγ (r3 )α C (r3 ,r3 ; iu)Q̂cγ (r3 ) du
(4)
This is the form of the three-body nonadditive dispersion
energy derived by Stogryn [18], which is valid for large-gap
systems only. If we retain only the dipole-dipole terms in
the Stogryn expression and make the further assumption that
we are dealing with systems of isotropic sites of (average)
a
a
polarizability ᾱ a , we can use ααα
δaa = ᾱ δαα , and we obtain
the Axilrod-Teller-Muto [19,20] triple-dipole term [19,20]
(3)
Edisp,MP
[3,ATM] =
abc
(3)
[3](loc)
Edisp,MP
=+
T ab T bc T ca
π (4π 0 )3 α β β γ γ α
∞
a
b
c
×
ααα
(iu)αββ (iu)αγ γ (iu)du.
0
aa
bb
cc
αα,α
(iu)αββ (iu)αγ γ (iu)du.
This is the generalized (distributed) multipole expansion for
the three-body nonadditive dispersion energy.
For systems with large HOMO–LUMO gaps (band gaps
in infinite systems), Misquitta et al. [2] have shown that
the nonlocal polarizabilities decay rapidly with intersite
separation. The characteristic decay length becomes smaller
as the gap increases. In this case, the nonlocal polarizabilities
can be localized using a multipole expansion [24,25] and we
aa a
can replace ααα
by a local equivalent ααα δaa in Eq. (4) to give
(5)
(3)
ab
where Tαβ
is the interaction function [5] between multipole
α on site a (in subsystem A) and multipole β on site b
(in subsystem B). At lowest order, the interaction function
ab
T00,00
= |ra − rb |−1 describes the interaction of the charge on
a with that on b. With this multipole expansion (MP), Eq. (1)
takes the form
0
α A (r1 ,r1 ; iu)α B (r2 ,r2 ; iu)α C (r3 ,r3 ; iu)
.
|r1 − r2 ||r2 − r3 ||r3 − r1 |
(3)
Edisp
[3]
PHYSICAL REVIEW B 89, 045140 (2014)
C9abc
1 + 3 cos â cos b̂ cos ĉ
, (6)
3
3 R3
(4π 0 )3 Rab
Rac
bc
where the C9abc dispersion coefficient is defined by
3 ∞ a
C9abc =
ᾱ (iu)ᾱ b (iu)ᾱ c (iu)du
π 0
045140-3
(7)
MISQUITTA, MAEZONO, DRUMMOND, STONE, AND NEEDS
III. COMPUTATIONAL DETAILS AND RESULTS
A.
(3)
[3]
Edisp
100
10−10
a ,β
Iγbc (iu) =
αγbbδ (iu)Tδcb ,
(8)
b ,δ
ca
Iα
(iu) =
cc
ac
αφ
(iu)Tαφ
.
d−8.94
−8.89
d
d−4.86
d−6.58
10−14
from nonlocal polarizabilities
The naı̈ve evaluation of Eq. (4) incurs a computational
cost that scales as O[n6 (l + 1)12 K], where n is the number
of sites, l is maximum rank of the polarizability matrix, and K
is the number of quadrature points, typically 10. The scaling
may be improved by calculating and storing the following
intermediates:
ab
aa ba Iαγ
(iu) =
ααβ
(iu)Tγβ
,
d−3.23
10−5
u3/N / a.u.
and â is the angle subtended at site a by unit vectors r̂ab
and r̂ac , with similar definitions for the angles b̂ and ĉ.
This is the more commonly used form of the nonadditive
dispersion energy, though, as we see from this derivation, like
the Stogryn expression, Eq. (6) is valid only for large-gap
systems (insulators).
PHYSICAL REVIEW B 89, 045140 (2014)
10−20
η = 2.0
η = 1.5
η = 1.0
d−8.80
10
d / a.u.
100
1000
FIG. 2. (Color online) The third-order nonadditive dispersion
energy calculated using the nonlocal charge-flow (rank-0) polarizabilities of (H2 )64 chains with bond-alternation parameters η = 1, 1.5,
and 2. The wires are parallel and arranged in an equilateral triangular
configuration with side d. Each set of data is associated with two
straight-line fits of the form ∼d −x to the data in the near (solid lines)
and far (dashed lines) regions. Broadly, the transition from the shortto long-range behavior is in the region of the intersection of these
lines.
c ,φ
The total computational cost of calculating these intermediates
is O[n3 (l + 1)6 K]. Equation (4) now takes the form
∞
(3)
ca
Edisp,MP [3] =
I ab (iu)Iγbc (iu)Iα
(iu)du
π (4π 0 )3 0 αγ
∞
ac
ca
=
Jα
(iu)Iα
(iu)du,
(9)
π (4π 0 )3 0
where we have defined yet another intermediate
ac
ab
(iu) =
Iαγ
(iu)Iγbc (iu),
Jα
(10)
b,γ
which incurs a computational cost of O[n3 (l + 1)6 K]. Equation (9) is evaluated with a computational cost of O[n2 (l +
1)4 K], so the overall cost of the calculation is only O[4n3 (l +
1)6 K], a significant improvement from the naive cost reported
above.
We have studied the interactions between two parallel
finite (H2 )64 chains with bond-alternation parameters η =
2.0, 1.5, and 1.0, where η is the ratio of the alternate
bond lengths. Frequency-dependent polarizability calculations
were performed with coupled Kohn-Sham perturbation theory
using the PBE functional and the adiabatic local density
approximation (LDA) linear-response kernel with the SadlejpVTZ basis set [26]. Calculations on shorter chains indicated
that the PBE results were qualitatively the same as those
from the more computationally demanding PBE0 functional.
The Kohn-Sham density functional theory (DFT) calculations
were performed using the NWCHEM program [27] and the
coupled Kohn-Sham perturbation theory, and polarizability
calculations were performed with the CAMCASP program [28].
Dispersion energies were calculated with the DISPERSION
program that is available upon request.
A finite hydrogen chain with bond-length alternation is
a convenient model for a 1D wire as we can control the
metallicity of the system using the alternation parameter η:
with η = 2.0, 1.5, and 1.0, the Kohn-Sham HOMO–LUMO
gap of the chain is 7.5, 3.1, and 1.6 eV, respectively, the
undistorted chain being the most metallic.
We have calculated distributed nonlocal polarizabilities
with terms from rank 0 (charge) to 4 (hexadecapole) using
a constrained density-fitting algorithm [23]. This technique
has been demonstrated to result in a compact and accurate
description of the frequency-dependent polarizabilities, with
relatively small charge-flow terms. Furthermore, Misquitta
et al. [2] have demonstrated that these polarizabilities can
accurately model the two-body dispersion energies between
hydrogen chains for which terms of rank 0 are sufficient; the
agreement with nonexpanded symmetry-adapted perturbation
(2)
theory based on density functional theory [SAPT(DFT)]Edisp
energies being excellent even for chain separations as small
as 6 a.u. We expect a similar accuracy for the three-body
nonadditive dispersion energy investigated in this paper.
(3)
[3] energies per H2 unit
In Figs. 2 and 3, we report Edisp,MP
for the equilateral triangular and coplanar configurations of
the (H2 )64 trimer. The broad features of these figures are as
follows:
(i) There is no single power law that fits the data. Instead,
we have two distinct regions: for separations much larger
than the chain length (much greater than 70–100 a.u.), the
nonadditive dispersion energy decays as ∼d −9 , consistent with
the Axilrod-Teller-Muto expression [Eq. (6)]. This is because
at such large separations the chains appear to each other as
point particles.
(ii) At sufficiently short separations, we see another powerlaw decay, but with an exponent that varies with the bond
alternation η of the wire. For the most insulating wire with
η = 2.0, the short-separation exponent is relatively close to 7,
the value expected from the summation of trimers of atoms,
045140-4
ANOMALOUS NONADDITIVE DISPERSION INTERACTIONS . . .
100
d−3.28
d−8.79
u3/N / a.u.
10−5
10−10
PHYSICAL REVIEW B 89, 045140 (2014)
d−8.64
d−5.05
d−8.48
d−6.69
10−14
10−20
η = 2.0
η = 1.5
η = 1.0
10
d / a.u.
100
1000
FIG. 3. (Color online) The third-order nonadditive dispersion
energy calculated using the nonlocal charge-flow (rank-0) polarizabilities of (H2 )64 chains with bond-alternation parameters η = 1,
1.5, and 2. The wires are parallel, coplanar, and equally spaced.
while for the most metallic wire with η = 1.0 the exponent is
close to 3.
(iii) The nonadditive dispersion energy is enhanced as the
degree of metallicity increases, and for the most metallic wires
is nearly four orders of magnitude larger than that for the most
insulating wire.
(iv) The charge-flow polarizabilities are responsible for
both the change in power-law exponent at short range and
the enhancement at long range. Contributions from nonlocal
dipole fluctuations, that is, terms of rank 1 (not shown in the
figures), are insignificant by comparison. This was also the
observation of Misquitta et al. [2] for the two-body dispersion
energy.
(v) The Axilrod-Teller-Muto triple-dipole expression leads
to a favorable three-body nonadditive dispersion energy for
three atoms in a linear configuration. However, for three wires
in such a configuration (Fig. 3), the nonadditivity is positive,
i.e., unfavorable.
These observations should perhaps not come as a surprise
as they are analogous to those obtained by Misquitta et al.
[2] for the two-body dispersion energy between 1D wires.
However, the deviations from the standard picture are much
more dramatic here. In going from the insulating η = 2.0 to
near-metallic wire, the two-body dispersion exhibits a largeseparation enhancement of two orders of magnitude compared
with four orders for the three-body nonadditive dispersion, and
for small wire separations the power law changes from d −5 to
d −2 for the two-body energy while it changes from d −7 to d −3
for the three-body nonadditivity.
In an analogous manner to the second-order dispersion
(2)
(3)
, the anomalous nature of Edisp
[3] can be explained
energy Edisp
using a simple charge-fluctuation picture. In Fig. 4, we
depict the plasmonlike long-range electronic fluctuations in
the wires arranged in the equilateral triangular geometry. The
dispersion interaction will be associated with both local and
extended fluctuations. The local fluctuations give rise to the
(3)
standard model for Edisp
[3]. Here, we are concerned with the
extended, plasmonlike fluctuations of typical length scale lc ,
as depicted in the figure. An extended + . . . − spontaneous
fluctuation in one wire induces a − . . . + fluctuation in the
FIG. 4. (Color online) The anomalous three-body nonadditive
dispersion interaction between three parallel 1D wires (in blue)
in an equilateral arrangement can be rationalized on the basis of
correlations in long-range fluctuations (red arrows). Here, d is the side
of the triangle and lc is the typical correlation length for electronic
fluctuations. The spontaneous and induced extended fluctuations are
indicated by the double-headed arrows, and their signs by the + . . . −
labels.
second, which, in turn, induces a + . . . − fluctuation in the
third. The interaction between the first and third will always
(3)
be repulsive leading to a positive Edisp
[3] energy. If the wire
separations satisfy d < lc , the extended fluctuations can not
be regarded as dipoles, instead, as shown in Fig. 4, their
interactions are modeled as between two trimers of charges
resulting from extended charge fluctuations. Each pair of
charges in a trimer interacts as d −1 , leading to an effective
three-body nonadditive dispersion of u3 ∼ +d −3 . For wire
separations much larger than lc , the extended fluctuations can
be modeled as dipoles. Each pair of these dipoles interacts as
±d −3 , giving rise to a +d −9 contribution to the nonadditive
dispersion energy. But, all such interactions must be summed
over, leading to the effective u3 ∼ +d −7 behavior. If the wires
are finite in extent, we recover the u3 ∼ +d −9 power law for
separations much larger than the wire length.
It is now well known that Kohn-Sham time-dependent
linear-response theory is not quantitatively accurate for heavily
delocalized systems, with polarizabilities typically overestimated [29–31], and hyperpolarizabilities even more so.
One may therefore question the validity of our calculations.
We seek, however, a description of the physical effect and
make no claims to being quantitatively accurate. We know
from the range of calculations described in the Introduction
that our hydrogen chain models are able to describe the physics
of the two-body dispersion energy between 1D wires, and we
see no reason to doubt their validity for trimers of such wires.
Nevertheless, to remove any possibility of doubt, we have used
QMC techniques to corroborate the results obtained with these
models.
B. Diffusion Monte Carlo (DMC) calculations
In our DMC calculations, we considered parallel biwires
and parallel triwires in an equilateral-triangle configuration
045140-5
MISQUITTA, MAEZONO, DRUMMOND, STONE, AND NEEDS
PHYSICAL REVIEW B 89, 045140 (2014)
with interwire spacing d. Each wire was modeled by a
single-component 1D HEG of density parameter rs in a
cell of length L(rs ,N) = 2N rs subject to periodic boundary
conditions, where N is the number of electrons per wire
in the cell. The electron-electron interaction was modeled
by a 1D Coulomb potential [32]. The charge neutrality of
each wire was maintained by introducing a uniform line
of positive background charge. To estimate the asymptotic
binding behavior between long, metallic wires we must have
results with a previous study [3], we used the same time steps:
0.04, 0.2, and 2.5 a.u. at rs = 1, 3, and 10, respectively. These
are sufficiently small that the time-step bias in our results is
negligible. Our QMC calculations were performed using the
CASINO code [36].
L(rs ,N) d rs .
(11)
We chose to work with real wave functions at the point of
the simulation-cell Brillouin zone, and the largest systems we
considered had N = 111 electrons per wire (333 electrons in
total for the triwire). To investigate finite-size errors, we also
performed calculations with N = 5, 11, 21, and 55 electrons
per wire.
We used many-body trial wave functions of Slater-Jastrow
backflow type. Each Slater determinant contained plane-wave
orbitals of the form exp(ikx). The use of single-component
(i.e., fully spin-polarized) HEGs is justified in Ref. [3]. DMC
calculations for strictly 1D systems do not suffer from a
fermion sign problem because the nodal surface is completely
defined by electron coalescence points, where the trial wave
function goes to zero. Our DMC calculations are therefore essentially exact for the systems studied, although these systems
are finite wires subject to periodic boundary conditions rather
than infinite wires. Electrons in different wires were treated as
distinguishable, so the triwire (biwire) wave function involves
the product of three (two) Slater determinants. Our Jastrow
exponent [33] was the sum of a two-body function consisting of
an expansion in powers of interelectron in-wire separation up
to 10th order, and a two-body function consisting of a Fourier
expansion with 14 independent reciprocal-lattice points. These
functions contained optimizable parameters whose values
were allowed to differ for intrawire and interwire electron
pairs.
We employed a backflow transformation in which the
electron coordinates in the Slater determinants were replaced
by “quasiparticle coordinates” that depend on the positions
of all the electrons. We used the two-body backflow function
of Ref. [34], which consists of an expansion in powers of
interelectron in-wire separation up to 10th order, again with
separate terms for intrawire and interwire electron pairs.
Backflow functions are normally used to improve the nodal
surfaces of Slater determinants in QMC trial wave functions
[34]. In the strictly 1D case, the backflow transformation leaves
the (already exact) nodal surface unchanged, but it provides a
compact parametrization of three-body correlations [35].
The values of the optimizable parameters in the Jastrow
factor and backflow function were determined within VMC
by minimizing the mean absolute deviation of the local
energy from the median local energy [36]. The optimizations
were performed using 32 000 statistically independent electron
configurations to obtain statistical estimators, while 3200 configurations were used to determine updates to the parameters
[37,38].
Our DMC calculations were performed with a target
population of 1280 configurations. The first 500 steps were
discarded as equilibration. To aid comparison of the present
C. DMC results
We denote the total energy of the N -electron M-wire system
as EM , and the total energy per electron as eM , so e1 = E1 /N .
The parallel two-wire system has an additional interaction
energy E2 (d), so the energy per electron is
e2 (d) = [2E1 + E2 (d)]/2N ≡ e1 + u2 (d),
(12)
consequently, the biwire interaction energy per electron u2 (d)
is
u2 (d) = e2 (d) − e1 .
(13)
Similarly, the equilateral-triangle configuration, parallel threewire system has an energy per electron of
e3 (d) = [3E1 + 3E2 (d) + E3 (d)]/3N
≡ e1 + 2u2 (d) + u3 (d),
(14)
from which we get the nonadditive contribution to the energy
of the triwire system per electron to be
u3 (d) = e3 (d) − e1 − 2u2 (d) = e3 (d) − 2e2 (d) + e1 . (15)
We fitted
exp(C)
,
(16)
dα
where C and α are fitting parameters to our DMC results for
|u2 (d)| and |u3 (d)| (extrapolated to the thermodynamic limit),
for d in the asymptotic regime. As shown in Figs. 5–7, the
asymptotic binding energies u2 (d) and u3 (d) show power-law
behavior as a function of d at all densities.
To estimate the finite-size errors at a given wire separation
d, we examined the variation of the energy with the number
N of electrons per wire. It has recently been reported [35] that
the finite-size error in the total energy per electron of the 1D
HEG scales as
c
(17)
e1 (N ) = e1 (∞) + 2 ,
N
where c is a constant, over the range of N considered here. Our
results for e2 and e3 , shown in Fig. 8, are consistent with this
dependence. However, we find that the interaction energies u2
and u3 at a given d show a more slowly decaying finite-size
error:
c
uM (N ) = uM (∞) + ,
(18)
N
where c is a constant. Hence, Eq. (17) can not give the
asymptotic form of the finite-size error in the total energy
of a 1D system in the limit of large N .
We have extrapolated the binding-energy data shown in
Figs. 5–7 to the thermodynamic limit at each d using Eq. (18).
We have then fitted Eq. (16) to the extrapolated binding-energy
data for triwires and biwires, respectively. The resulting fitting
parameters, including the asymptotic exponents, are given in
Table I.
045140-6
u(d) =
ANOMALOUS NONADDITIVE DISPERSION INTERACTIONS . . .
PHYSICAL REVIEW B 89, 045140 (2014)
−10−3
d−2.310(1)
d−2.5410(7)
u2 / a.u.
u2 / a.u.
−10−3
−4
−10
N =∞
N = 111
N = 55
Drummond et al. N = 101
Drummond et al. N = 55
−10−5
1
d / a.u.
−10−4
N =∞
N = 111
N = 55
Drummond et al. N = 101
Drummond et al. N = 55
−10−5
10
d / a.u.
10
−3
10
d−2.670(5)
10−4
u3 / a.u.
u3 / a.u.
10−4
10−5
10−6
−7
10
N =∞
N = 111
N = 55
1
d−2.435(8)
d / a.u.
10−5
N =∞
N = 111
N = 55
−6
10
d / a.u.
10
FIG. 5. (Color online) DMC results for the asymptotic behavior
of the biwire interaction u2 (left panel) and the nonadditive triwire
contribution u3 (right panel) at rs = 1.
FIG. 6. (Color online) DMC results for the asymptotic behavior
of the biwire interaction u2 (left panel) and the nonadditive contribution u3 (right panel) at rs = 3.
−10−3
IV. DISCUSSION
d−2.649(2)
u2 / a.u.
−10−4
−10−5
N =∞
N = 111
N = 55
Drummond et al. N = 75
Drummond et al. N = 55
−6
−10
−10−7
10
100
d / a.u.
10−3
10−4
u3 / a.u.
We have investigated the nature of the nonadditive dispersion between three parallel wires and we have demonstrated
that as the HOMO–LUMO gap (band gap in infinite wires)
(3)
[3] from the conventional
decreases, the deviations of Edisp
triple-dipole Axilrod-Teller-Muto model increase. These deviations occur mainly in two ways:
(i) For wire separations smaller than the typical electron correlation length, the effective three-body nonadditive
dispersion behaves as u3 (d) ∼ d −β , where β → 3 as the
HOMO–LUMO gag decreases. This power-law arises from
the correlations between extended charge fluctuations that
are associated from the plasmonlike modes in the wires.
This is a substantially slower decay than the u3 (d) ∼ d −7
behavior expected from the standard triple-dipole summations
associated with local dipole fluctuations. For finite wires,
u3 (d) ∼ d −9 for separations much larger than the wire length.
(ii) u3 (d) is substantially enhanced as the gap reduces. This
is most dramatic for large separations, where we observed an
enhancement of four orders of magnitude for the near-metallic
wires compared with the wires with the largest HOMO–
LUMO gap.
These observations are analogous to those obtained by Misquitta et al. [2] with regard to the second-order dispersion
(2)
energy Edisp
, although the effects of metallicity are more
dramatic for the three-body nonadditivity. We have provided
a simple physical picture of correlations in extended charge
101
d−2.88(2)
10−5
10−6
10−7
10−8 10
N =∞
N = 111
N = 55
d / a.u.
100
FIG. 7. (Color online) DMC results for the asymptotic behavior
of the biwire interaction u2 (left panel) and the nonadditive contribution u3 (right panel) at rs = 10.
045140-7
MISQUITTA, MAEZONO, DRUMMOND, STONE, AND NEEDS
PHYSICAL REVIEW B 89, 045140 (2014)
TABLE I. Values of power-law parameters in Eq. (16) for the
two-body and three-body energies.
-0.1428
-0.1429
u2 < 0
(hartree/wire)
-0.1430
-0.1431
rs = 1
rs = 3
rs = 10
-0.1432
e2
-0.1433
u3 > 0
C
α
C
α
− 6.0685(6)
− 4.084(1)
− 2.114(6)
2.310(1)
2.5410(7)
2.649(2)
− 7.942(5)
− 5.565(8)
− 2.98(5)
2.435(8)
2.670(5)
2.88(2)
-0.1434
-0.1435
fluctuations using which both of these observations can be
understood.
We have established these results using two techniques:
(3)
[3] that includes
(1) a generalized multipole expansion for Edisp
contributions from charge-flow polarizabilities responsible for
the long-wavelength, plasmonlike fluctuations, and (2) DMC.
The former has the advantage that we can directly calculate
(3)
Edisp
[3], but it is applicable only to finite systems with nonzero
HOMO–LUMO gaps. By contrast, DMC is applicable to
infinite systems (modeled in cells subject to periodic boundary
conditions) with zero gaps, and in principle is able to describe
the third-order correlation energy exactly. However, like any
supermolecular technique, that is, techniques that calculate
the interaction energy from total-energy differences, DMC
(3)
is unable to separate the two-body energy Edisp
[2] from
-0.1436
0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035 0.040
-2
N
-0.1428
-0.1429
(hartree/wire)
-0.1430
-0.1431
-0.1432
e3
-0.1433
-0.1434
-0.1435
-0.1436
0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035 0.040
-2
N
-6
-2.0×10
u2
(hartree/wire)
-4.0×10-6
-6.0×10-6
-8.0×10-6
-1.0×10-5
-5
-1.2×10
-1.4×10-5
-1.6×10-5
0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20
N-1
3.0×10-6
u3
(hartree/wire)
2.5×10-6
2.0×10-6
1.5×10-6
1.0×10
-6
5.0×10-7
0.0×10
0
-5.0×10-7
0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20
N-1
FIG. 8. (Color online) DMC results for the N dependence of the
total biwire (e2 ) and triwire (e3 ) energies and interaction energies (u2
and u3 ) at rs = 10 and at interwire spacing d = 30 a.u. The data at
N = 5 (1/N = 0.2, 1/N 2 = 0.04) were excluded from the fits (solid
lines).
(3)
the three-body nonadditive dispersion Edisp
[3]. Nevertheless,
there is a consistency in the results from these two methods. At
short range (i.e., at separations less than the correlation length)
the multipole expansion used on trimers of finite (H2 )64 chains
yields a power law of u3 (d) ∼ d −β where β → 3+ , that is, β
approaches 3 from above, while in the DMC results, β → 3−
as rs increases. For small rs , the exponent is significantly
smaller than 3. This could be because of finite-size effects,
(3)
contributions from Edisp
[2], or it could be a genuine effect not
captured by the multipole expansion.
The increased effect of the plasmonlike, charge-flow
(3)
(2)
fluctuations on Edisp
[3] compared with Edisp
is related to the
long range of these fields produced by the fluctuations. The
dipole fluctuations in insulators result in electric fields that
behave as r −3 , a rapid decay compared with the r −1 behavior
of the electric fields from the plasmon-type fluctuations.
Consequently, we expect the many-body expansion to be
slowly convergent for conglomerates of low-dimensional
semimetallic systems. As we have demonstrated, the threebody nonadditivity quenches the already enhanced two-body
dispersion. Likewise, by extending our physical model for
these anomalous dispersion effects, we expect that the fourbody nonadditivity will be attractive and decay as −d −4 for 1D
metallic systems, and will consequently quench the three-body
nonadditivity.
The slow decay and alternating signs of the N -body
nonadditive dispersion suggests that the many-body expansion
may not be a useful way of modeling the dispersion interaction
in, say, a bundle of 1D semimetallic wires. An alternative may
be a generalization of the self-consistent polarization model
proposed by Silberstein [39] and Applequist [40], and recently
significantly developed by Tkatchenko et al. [41]. However,
models such as these would have to be modified to include the
045140-8
ANOMALOUS NONADDITIVE DISPERSION INTERACTIONS . . .
PHYSICAL REVIEW B 89, 045140 (2014)
charge-flow polarizabilities to be able to describe the metallic
effects described in this paper.
For finite molecular systems, the changes in power law
described here are, to an extent, of academic interest only.
In practice, subtle power-law changes in the dispersion
interaction can be easily masked by the other, often larger,
components of the interaction energy, particularly the firstorder electrostatic energy. While this may be the case, it
is the second effect, the enhancement of the dispersion
energy that arises from the plasmonlike modes, that may
have a perceptible effect. The long-wavelength fluctuations
cause an enhancement of the effective two- and three-body
dispersion coefficients. We believe that this effect, which is
captured by techniques such as the Williams-Stone-Misquitta
method [14,15], may prove significant even for relatively small
molecular systems. We are currently working to investigate this
phenomenon.
[1] J. Spencer, Ph.D. thesis, St. Catharine’s College, University of
Cambridge, 2009.
[2] A. J. Misquitta, J. Spencer, A. J. Stone, and A. Alavi, Phys. Rev.
B 82, 075312 (2010).
[3] N. D. Drummond and R. J. Needs, Phys. Rev. Lett. 99, 166401
(2007).
[4] J. F. Dobson, A. White, and A. Rubio, Phys. Rev. Lett. 96,
073201 (2006).
[5] A. J. Stone, The Theory of Intermolecular Forces, 2nd ed.
(Oxford University Press, Oxford, 2013).
[6] I. G. Kaplan, Intermolecular Interactions, 2nd ed. (Wiley, New
York, 2005).
[7] W. M. C. Foulkes, L. Mitas, R. J. Needs, and G. Rajagopal, Rev.
Mod. Phys. 73, 33 (2001).
[8] J. F. Dobson, Surf. Sci. 601, 5667 (2007).
[9] J. F. Dobson, K. McLennan, A. Rubio, J. Wang, T. Gould,
H. M. Le, and B. P. Dinte, Aust. J. Chem. 54, 513 (2001).
[10] C. A. Coulson and P. L. Davies, Trans. Faraday Soc. 48, 777
(1952).
[11] H. C. Longuet-Higgins and L. Salem, Proc. R. Soc. A 259, 433
(1961).
[12] D. B. Chang, R. L. Cooper, J. E. Drummond, and A. C. Young,
Phys. Lett. A 37, 311 (1971).
[13] V. V. Gobre and A. Tkatchenko, Nat. Commun. 4, 2341
(2013).
[14] A. J. Misquitta and A. J. Stone, J. Chem. Theory Comput. 4, 7
(2008).
[15] A. J. Misquitta, A. J. Stone, and S. L. Price, J. Chem. Theory
Comput. 4, 19 (2008).
[16] J. G. Angyan, Int. J. Quantum Chem. 109, 2340 (2009).
[17] V. A. Parsegian, Van der Waals Forces: A Handbook for
Biologists, Chemists, Engineers, and Physicists (Cambridge
University Press, Cambridge, UK, 2005).
[18] D. E. Stogryn, Mol. Phys. 22, 81 (1971).
[19] P. M. Axilrod and E. Teller, J. Chem. Phys. 11, 299 (1943).
[20] Y. Muto, Jour. Phys.-Math. Soc. Japan 17, 629 (1943).
[21] H. C. Longuet-Higgins, Disc. Faraday Soc. 40, 7 (1965).
[22] A. J. Stone, Mol. Phys. 56, 1065 (1985).
[23] A. J. Misquitta and A. J. Stone, J. Chem. Phys. 124, 024111
(2006).
[24] C. R. Le Sueur and A. J. Stone, Mol. Phys. 83, 293 (1994).
[25] T. C. Lillestolen and R. J. Wheatley, J. Phys. Chem. A 111,
11141 (2007).
[26] A. J. Sadlej, Collect. Czech Chem. Commun. 53, 1995 (1988).
[27] E. J. Bylaska, W. A. de Jong, K. Kowalski, T. P. Straatsma,
M. Valiev, D. Wang, E. Apra, T. L. Windus, S. Hirata, M. T.
Hackler et al., NWCHEM, a computational chemistry package for
parallel computers, version 5.0.
[28] A. J. Misquitta and A. J. Stone, CAMCASP, a program for studying intermolecular interactions and for the
calculation of molecular properties in distributed form,
http://www-stone.ch.cam.ac.uk/programs.html#CamCASP.
[29] B. Champagne, E. A. Perpete, S. J. A. van Gisbergen, E.-J.
Baerends, J. G. Snijders, C. Soubra-Ghaoui, K. A. Robins, and
B. Kirtman, J. Chem. Phys. 109, 10489 (1998).
[30] B. Champagne, D. H. Mosley, M. Vracko, and J.-M. Andre,
Phys. Rev. A 52, 178 (1995).
[31] B. Champagne, D. H. Mosley, M. Vracko, and J.-M. Andre,
Phys. Rev. A 52, 1039 (1995).
[32] V. R. Saunders, C. Freyria-Fava, R. Dovesi, and C. Roetti, Comp.
Phys. Commun. 84, 156 (1994).
[33] N. D. Drummond, M. D. Towler, and R. J. Needs, Phys. Rev. B
70, 235119(11) (2004).
[34] P. Lopez Rios, A. Ma, N. D. Drummond, M. D. Towler, and
R. J. Needs, Phys. Rev. E 74, 066701 (2006).
[35] R. M. Lee and N. D. Drummond, Phys. Rev. B 83, 245114
(2011).
[36] R. J. Needs, M. D. Towler, N. D. Drummond, and P. L. Rios,
J. Phys.: Condens. Matter 22, 023201 (2010).
[37] J. R. Trail, Phys. Rev. E 77, 016703 (2008).
[38] J. R. Trail and R. Maezono, J. Chem. Phys. 133, 174120 (2010).
[39] L. Silberstein, Philos. Mag. 33, 92 (1917).
[40] J. Applequist, J. R. Carl, and K.-K. Fung, J. Am. Chem. Soc.
94, 2952 (1972).
[41] A. Tkatchenko, R. A. DiStasio, R. Car, and M. Scheffler, Phys.
Rev. Lett. 108, 236402 (2012).
ACKNOWLEDGMENTS
Financial support was provided by the U. K. Engineering
and Physical Sciences Research Council (EPSRC). Part of the
computations have been performed using the K computer at
Advanced Institute for Computational Science, RIKEN. R.M.
is grateful for financial support from KAKENHI Grants No.
23104714, No. 22104011, and No. 25600156, and from the
Tokuyama Science Foundation.
045140-9